scholarly journals The motion of a rotor carried by a flexible shaft in flexible bearings

1933 ◽  
Vol 142 (846) ◽  
pp. 92-118 ◽  
Keyword(s):  
Equilibrium State ◽  
Critical Speed ◽  
Transverse Motion ◽  
Initial Displacement ◽  
Flexible Shaft ◽  
Great Amplitude ◽  
Set Up

1.1.— Scope of Investigation .—This paper discusses the transverse motion of a rotor carried by a flexible shaft rotating in flexibly-supported bearings. The rotor is assumed to consist of one or more rigid bodied mounted on a shaft which is weightless and torsionally rigid. The rotor and shaft are in rotation; in the first place, it is taken that the speed of rotation is maintained constant, driving torques being applied if necessary about the shaft axis. Unsymmetrical flexibility of the bearing supports and unsymmetrical transverse flexibility of the shaft of the flexible members of the system is also considered. The work is mainly analytical, but reference is also made to experiments which have been carried out. 1.2. Definitions . —The flexible members (shaft and bearing supports) are described collectively as the mounting. An unstable speed is a speed of rotation at which a rotor, after receiving a small initial displacement from its equilibrium state of motion, tends to increase its amplitude of vibration beyond all bounds; while a critical speed is a speed at which out-of-balance alone tends to set up vibrations of very great amplitude.

Bifurcation and Chaos Behavior of Rotor-Bearing System With Crack and Pedestal Looseness Faults

2007 ◽  
Author(s):  
Yuegang Luo ◽  
Songhe Zhang ◽  
Feng Wen ◽  
Bangchun Wen
Keyword(s):  
Critical Speed ◽  
Journal Bearings ◽  
Cracked Rotor ◽  
Bifurcation And Chaos ◽  
Coupling Faults ◽  
Speed Range ◽  
Rotor Bearing ◽  
Set Up ◽  
Bearing System ◽  
Main Influence

A dynamic model was set up for the two-span rotor-bearing system with coupling faults of crack and pedestal looseness supported on three plain journal bearings. The nonlinear dynamic behaviors that induced by crack, pedestal looseness and coupling faults are numerically studied. There is quasi-periodic motion appearing in the cracked rotor-bearing system, and it within the sub-critical speed range in the pedestal looseness rotor-bearing system. There is chaotic motion appearing within the supper-critical speed range in the pedestal looseness rotor-bearing system. The pedestal looseness fault is the main influence on the coupling faults system, and there is Period-3 motion appearing in the system. The results may bring up theoretical references for fault diagnoses, dynamic design, and security running to rotor-bearing system.

A Simple Approach to the Wing Flutter Problem

1933 ◽  
Vol 37 (273) ◽  
pp. 783-792 ◽  
Author(s):  
B. Lockspeiser
Keyword(s):  
Elastic Energy ◽  
Simple Approach ◽  
Inertial Forces ◽  
Initial Displacement ◽  
Wing Flutter ◽  
Wing Section ◽  
Air Damping ◽  
Gain Energy ◽  
Wing Structure ◽  
Set Up

Let us first consider the oscillations, in still air, of a monoplane wing whose aileron is supposed locked to the wing in such a way that it behaves as though it were an integral part of the wing structure. When the wing is displaced from its position of equilibrium and released it will, in general, vibrate both in flexure and torsion. The initial displacement may be purely flexural, but if the inertial forces called into play, over any wing section, produce a twisting moment about the centre of twist (i.e., the centre about which the wing section twists on the application of a pure torque at that section) torsional as well as flexural oscillations will be set up. Inertia, in general, robs the two kinds of oscillation of their independence, and, when they are interdependent, we may conveniently speak of “inertial couplings” between the two motions. In still air these vibrations must, of necessity, die down. One part of the wing may gain energy at the expense of another, but the store of elastic energy given to the wing by the initial displacement must grow progressively less as the wing does work against the viscous air damping and structural hysteresis forces.

A Study of Shaft Vibration Based on Transfer Matrix

2013 ◽  
Vol 365-366 ◽  
pp. 339-343
Author(s):  
Chang Tan ◽  
Zhi Ling Guo ◽  
Rui Kun Zhou
Keyword(s):  
Numerical Simulation ◽  
Transfer Matrix ◽  
Critical Speed ◽  
Lumped Parameter Model ◽  
Lumped Parameter ◽  
The Common ◽  
Shaft Vibration ◽  
Set Up

This paper first analyze the method of transfer matrix, set up the lumped parameter model. then figure out the common transfer matrix of shaft. Take some shaft as an example, using matlab calculate the critical speed and analyze the result. The analysis can provide basis and method for shaft vibration numerical simulation.

Critical Speed for Floppy Disks

1998 ◽  
Vol 65 (1) ◽  
pp. 116-120 ◽  
Author(s):  
A. A. Renshaw
Keyword(s):  
Thin Films ◽  
Free Vibration ◽  
Critical Speed ◽  
Transverse Motion ◽  
Design Criteria ◽  
Stability Prediction ◽  
Spinning Disk ◽  
Hydrodynamic Coupling ◽  
The Stability ◽  
Conventional Computer

This paper uses Lyapunov’s method to determine the critical speed of a flexible spinning disk enclosed in a housing that hydrodynamically couples the transverse motion of the disk to the motion of the thin films of air surrounding the disk. Depending on the clamping ratio, this critical speed is three to ten times higher than the critical speed in the absence of hydrodynamic coupling and does not depend on the strength of the hydrodynamic coupling. Despite the nonlinearity of the underlying model, the critical speed problem is linear and tractable. The linearized free-vibration problem is also computed to verify the stability prediction and to examine linearized damping and stiffness as possible design criteria. The results are relevant to the design of both conventional computer floppy disks and the emerging generation of 100+ MB floppies.

Optimal Control of the Rotor System Motion

2013 ◽  
Vol 705 ◽  
pp. 546-552
Author(s):  
Imankul Toleukhan
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Electric Drive ◽  
Rotor System ◽  
Flexible Shaft ◽  
Experimental Works ◽  
Set Up ◽  
Automatic Balancing ◽  
The Mathematical Model

Among the problems of the rotor machines dynamics the special attention is given to the problems of creation of the automatic balancing devices (ABD) in form of a hollow rotor, filled by a liquid, and the liquid-solidbody ABD. The theoretical and experimental works on research of the ABD on the base of a hollow rotor filled partially with a liquid and of the liquid-solidbody ABD are not enough. Therefore development of the methods of research of dynamics of the rotor machines with the ABD and such machines designs is an actual, new and perspective problem. In the present work the mathematical model of the rotor system with the ABD taking into account of the engine characteristics is offered. Lets consider the model of the rotor with electric drive with one disk, set up at the flexible shaft without skew. The shaft is lean on two bearings (fig. 1).

scholarly journals Characteristic Analysis of the Peak Braking Force and the Critical Speed of Eddy Current Braking in a High-Speed Maglev

Energies ◽  
2019 ◽  
Vol 12 (13) ◽  
pp. 2622
Author(s):  
Chuntao Chen ◽  
Jie Xu ◽  
Xibo Yuan ◽  
Xinzhen Wu
Keyword(s):  
Eddy Current ◽  
High Speed ◽  
Critical Speed ◽  
Analytical Formula ◽  
Analytical Calculation ◽  
Characteristic Analysis ◽  
Element Analysis ◽  
Braking System ◽  
Braking Force ◽  
Set Up

In the eddy current braking system of high-speed maglev, the peak braking force and the critical speed are key factors determining the performance of eddy current braking force. In this paper, the analytical formula of eddy current braking force is derived by a subdomain method considering the skin effect of the induction plate, and, subsequently, the characteristics of peak braking force and critical speed are analyzed. The analytical model is set up in a 2D Cartesian coordinate system. The Poisson equations in each subdomain are listed by treating the vector magnetic potential as a variable. By combining the boundary conditions between two adjacent subdomains, the expressions of eddy current density and magnetic density in the induction plate are obtained. Then, the analytical formula of the eddy current braking force is obtained by the Ampere force formula. The results of finite-element analysis confirm the validity of the analytical calculation. The methods of improving the performance of eddy current braking force under high speed are proposed by parametric analysis of peak braking force and critical speed, which provides guidance for the design of the eddy current braking system in high-speed maglev.

Dynamic Analysis and Research of Motor’s Elastic Shaft of the Large Miner High Shearer’s Cutting Parts

2011 ◽  
Vol 108 ◽  
pp. 105-110 ◽  
Author(s):  
Hong Bing Gao ◽  
Zhao Jian Yang
Keyword(s):  
Dynamic Analysis ◽  
Theoretical Basis ◽  
Critical Speed ◽  
Lagrange Equation ◽  
Angular Displacement ◽  
Analysis Model ◽  
Lumped Mass ◽  
Shaft System ◽  
Elastic Shaft ◽  
Set Up

Take one large miner high electric draught shearer for example. According to the lumped mass method, the gear’s moment of inertia of the shearer’s cutting part was equivalent to the gears connecting the output shaft of the electric cutting motor, the stiffness of central shaft and coupling was equivalent to the output elastic shaft of the electric cutting motor, and the dynamic analysis model of the elastic shaft of the cutting part was set up. Based on this model, the lagrange equation was built. The critical speed of the elastic shaft system, the motion situation on main vibration and harmonic vibration mode, motor shaft’s angular displacement, angular velocity, and angular acceleration’s dynamic response process under the condition of step loads were all analyzed and obtained. The critical speed provided theoretical basis for choosing the working speed of every shaft system of the shearer. The hollow elastic shaft had a protective effect to cutting motor’s damping and overload. The research results provided theoretical basis for the design of the elastic shaft of large miner high shearer.

Research on Vibration Characteristics of the Three-Bearings Supporting Structure with Cracks in the Rotor of the 25Cr2Ni4MoV Exciter

2019 ◽  
Vol 893 ◽  
pp. 33-38
Author(s):  
Qing Meng Zeng ◽  
Zhi Min Liu ◽  
You Liang Chen ◽  
Fan Yang Meng
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Critical Speed ◽  
Case Analysis ◽  
Element Model ◽  
Vibration Characteristics ◽  
Supporting Structure ◽  
Response Characteristics ◽  
Model Based ◽  
Set Up

Finite element model based on a unit with cracks in the rotor of the exciter is set up. Andthen critical speed is calculated that compared to actual measured value to verify the rationality of themodel. Lastly response characteristics of the three-bearings supporting structure are studied when thefirst critical speed of the exciter with cracks is closed to the working speed. And the reliability ofconclusions is further verified by case analysis.

Dynamic Characteristics of Nonlinear Elastics Rotor-Bearing System With Coupling Faults of Pedestal Looseness and Rub-Impact

2009 ◽  
Author(s):  
Yuegang Luo ◽  
Songhe Zhang ◽  
Zhaohui Ren ◽  
Bangchun Wen
Keyword(s):  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Critical Speed ◽  
Dynamic Design ◽  
Coupling Faults ◽  
Nonlinear Motion ◽  
Speed Range ◽  
Rotor Bearing ◽  
Set Up ◽  
Bearing System

A dynamic model of the nonlinear elastics rotor-bearing system with coupling faults of pedestal looseness and rub-impact was set up, taking the linearity and cube item as the physics nonlinear factors. The complex characteristics of the rotor-bearing system were numerically studied. There exists complex nonlinear motion of periodic, quasi-periodic and chaotic in the response of the system. The main motions of the rotor-bearing system with rub-impact fault are periodic-2, periodic-4 and quasi-periodic within the super-critical speed range, but it with coupling faults of pedestal looseness and rub-impact are periodic-3 and chaotic. The influence of oil-film force to the rotor system is weakened by the pedestal looseness fault. The results may bring up theoretical references for fault diagnoses, dynamic design, and security running to rotor-bearing system.

Should a Flexible Rotor Be Balanced in N or (N + 2) Planes?

1972 ◽  
Vol 94 (2) ◽  
pp. 548-558 ◽  
Author(s):  
W. Kellenberger
Keyword(s):  
Critical Speed ◽  
System Of Equations ◽  
Flexible Rotor ◽  
Orthogonal Functions ◽  
Linear System Of Equations ◽  
Flexible Rotors ◽  
Speed Range ◽  
Set Up ◽  
Limiting Case ◽  
Analytical Expressions

The problem of balancing flexible rotors consists mainly of eliminating rotating bearing forces. Analytical expressions are derived for the deformation and the rotating bearing forces of a rotor, using orthogonal functions. With this kind of representation it is possible to set up simple conditions for the vanishing rotating bearing forces. They lead to a linear system of equations giving the compensating unbalances in each of a set number of balancing planes. Two methods used in practice are theoretically explained and compared. The “N” method employs N planes for balancing a speed range up to, and including, the Nth critical speed and can be characterized by the condition A = 0, see equation (13). The “(N + 2)” method requires two more planes for the same speed range and is characterized by A = 0 and B = 0. It is proved that limlimN→∞B=0, so that in the limiting case of an infinite number of balancing planes (speed range from zero to infinity) both methods are of equal value. The two methods differ for finite N in their accuracy and the amount of calculation. Considering simple examples with known unbalance distribution it will be shown that the main error of the N method is the result of treating B as equal to 0, which it is not, thus accounting for the greater accuracy of the N + 2 method. The additional effort needed for the latter method is justified in those cases where greater accuracy is demanded.

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